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The Normal Distribution BUSA 2100, Sections 3.3, 6.2

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Introduction to the Normal Distribution l The normal distribution is the most widely used probability distribution. l Reason: Most variables observed in nature and many variables in business are “normally distributed.” l The normal distribution is a bell-shaped curve.

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Characteristics of the Normal Distribution l The normal distribution has 3 major characteristics. l (1) Most important characteristic -- Large frequencies near the mean, and small frequencies at the extremes. l (2) It is symmetric about the mean. l (3) It is infinite in extent (in theory) -- doesn’t touch the x-axis.

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Examples of the Normal Distribution l Women’s heights, men’s weights, IQs, daily sales. (Explain) l The size of the standard deviation affects the shape of the normal curve. l For all normal curves: 68+% of values are within +-1 std.dev.; 95+% of values are within +-2 std. dev.; 99+% of values are within +-3 std. dev.

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z-Values l Definition: A z-value represents the number of standard deviations that an item is from the mean. l A normal distribution has mean (mu) = 150 & std. dev. (sigma) = 20. Find the z-values for X = 170, 150, 130, & 195. l What is the z-value for X = 163? What is the formula for z?

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z-Values, Page 2 l A negative z-value means that the item is to the left of the mean. l Items with z-values beyond +-3 std. deviations are called outliers. l z-values, together with a normal curve table, can be used to find probabilities.

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Procedure for Calculating Normal Curve Probabilities l Step 1: Draw a sketch. l Step 2: Calculate the z-value(s). l Step 3: Look up the table value(s) in a normal curve table. l Step 4: Calculate the final answer.

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Normal Curve Example l Example 1: Suppose that daily sales for a product are normally distributed with mean 220 and standard deviation 36. l What is the approx. range for sales? l (a) What is the prob. that sales are less than 268?

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Normal Curve Example, p. 2 l (b) What is P(190 <= X <= 240)?

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Normal Curve Example, p. 3 l (d) What is the probability that sales are larger than 265? l The normal curve table measures all probabilities (areas) from the lower end.

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Types of Normal Curve Problems l All the problems we have just done are called regular normal curve problems: l Now we will do a backwards normal curve problem:

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Backwards Normal Curve Ex. l Ex. Mileage for a tire is normally distributed with mean 36,500 and std. dev. 5,000. l A customer refund will be given for the 10% of tires that get the least mileage. l What mileage (X-value) qualifies?

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Backwards Normal Curve Example, Page 2 l Solve z = (X - mu) / sigma for X.

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